each other in a remarkable degree, that incline us to choose the latter and thus,

When we say that light is polarized in a particular plane, we mean that the vibration of every particle is perpendicular to that plane.

Thus, in the undulation constituting the Ordinary ray of Iceland spar, the vibration of every particle is perpendicular to the principal plane of the crystal: in that constituting the Extraordinary ray, the vibration of every particle is parallel to the principal plane. When light falls upon unsilvered glass at the polarizing angle, the reflected wave is formed entirely by vibrations perpendicular to the plane of incidence: the transmitted wave is formed by some vibrations perpendicular to the plane of incidence, with an excess of vibrations parallel to the plane of incidence.

101. The reader will perceive that it is absolutely necessary to suppose, either that there are no vibrations in the direction of the wave's motion, or that they make no impression on the eye. For if there were such, there ought in the experiment of (98) to be visible fringes of interferences: of such however there is not the smallest trace.

102. As we now suppose light generally to consist of two sets of vibrations which cannot interfere with each other, it becomes important to establish some measure of the intensity of the compound light. It seems that this cannot be any other than the sum of the intensities corresponding to the two sets of vibrations. So that if the displacement from one vibration be represented by a sin (vt - x+4), and that from the other by b sin (vtx+B), the intensity of the mixed light will be a+b2. This then is the expression which we ought in strictness to have used in our former investigations. But as in all these (except those relating to reflection from plane glasses and lenses) the quantities a and b have in every part of the operation the same proportion, it is evident that the results, considered as giving the proportion of intensities of light, are in every instance correct.

PROP. 21. To explain on mechanical principles the trans

mission of a wave in which the vibrations are transverse to the direction of its motion.

103. In fig. 25 let the faint dots represent the original situations of the particles of a medium, arranged regularly in square order, each line being at the distance h from the next. Suppose all the particles in each vertical line disturbed vertically by the same quantity; the disturbances of different vertical lines being different. Let x be the horizontal abscissa of the second row; x-h that of the first, and x+h that of the third: let u, u,, and u' be the corresponding disturbances. The motions will depend upon the extent to which we suppose the forces are sensible. Suppose the only particles whose forces on A are sensible, to be

B, C, D, E, F, G,

(omitting those in the same line, as their attractions are equal and in opposite directions): and suppose them to be attractive, and as the inverse square of the distance: and the absolute force of each = m. The whole force tending to pull A downwards is

m (h+ u — u)

+

m (u — u,)

m (h

- u+u) {h2 + (h + u − u ̧)2 } * * {h2 + (u — u ̧)*j* ̄ ̄ {h2 + (h−u+u)2} 1

+

+

{h2 + (h + u — u')3}#* ̄ ̄ {h2 + (u — u')*}* ̄ ̄ {h2

{h2

Expanding these fractions, and neglecting powers of u and u-u above the first, the force tending to diminish u is

u,

and for u',

u +

du d2u h

h+

dx dx2 2'

an equation of exactly the same form as that for the transmission of sound (10). The solution therefore has the same form and therefore the transversal motion of particles sup

:

* If h is so large with regard to the length of a wave that the terms after h2 cannot be safely neglected, we may, by assuming a form for the function expressing u, integrate the equation

This expression increases as λ increases; that is, undulations consisting of long waves travel with greater velocity than those consisting of short waves. Thus the different refrangibility of differently coloured rays is accounted for. See Article 38. For other modifications of this theory, and their comparison with the observed indices of refraction of different rays in different media, the reader is referred to Professor Powell's treatise On the Undulatory Theory as applied to the explanation of unequal refrangibility.

posed here follows the same law as the direct motion of the particles of air: that is, it follows the law of undulation.

104. It seems probable that if we had supposed any other regular arrangement, or taken any other law of force, the same conclusion would have been obtained. And if we suppose the arrangement symmetrical with respect to certain fixed lines, but different in distance of particles, &c. in different directions (as for instance if we suppose every eight adjacent particles to be at the angles of a parallelopiped as in fig. 26), then for vibrations of the particles in different d'u directions the multiplier of will be different, and conse

dx2

quently the velocity of transmission of the wave (which is the square root of the multiplier) will be different. And the velocities of two waves may be different even when they are going in the same direction, provided that one of these waves consist of vibrations in one direction, and the other of vibrations in another direction, as if for instance in fig. 26 the directions of both waves were perpendicular to the paper, but if one set of vibrations were in the direction up and down, and the other in the direction right and left. For the force with which the particles act on each other depends on the distance of the particles in the direction of the waves' motion, and on their distance in the direction of the particles' vibration and in the case supposed, the latter element is different for the two waves, though the former is the same.

:

105. If the displacement of a particle, considered as in any direction, be resolved into three displacements in the directions of x, y, z, the variations of force in those directions produced by the alteration of a single particle (and consequently the force produced by the whole system) are the same as if the displacements in those directions had been made independently. From this it easily follows that the sum of any number of displacements causes forces equal to the sum of the forces corresponding to the separate displacements: and then, by the reasoning in (10) and (11), any number of undulations, produced by vibrations in different directions, may co-exist without disturbing each other.

PROP. 22. To explain the separation of common light into

two pencils by doubly-refracting crystals: and to account for the polarization of the two rays in planes at right angles to each other.

106. We shall assume for the state of the particles of ether within a crystal, an arrangement similar to that described in (104), or at least possessing this property, that there are three directions at right angles to each other, in which if a particle be disturbed, the resultant of the forces acting on it will tend to move it back in the same line in which the displacement is produced. These lines we suppose to be parallel to some lines determined by the form of the crystal.

107. Now in general the displacement+ of a particle or a series of particles will not produce a force whose direction coincides with the line of displacement. For suppose the disturbance in the direction of x to be X; that in the direction of y to be Y: and suppose the corresponding forces to be a X and by. The tangent of the angle made by the resultant b' Y force with the axis of x is but the tangent of the a2X angle made by the direction of displacement with the axis

Y

of x is : and these are different if a2 and l2 are different. X

In the same manner if we supposed a displacement Z in the direction of z, and if it produced a force cZ, the tangents of the angles, made by the projection of the resultant's direction

* M. Fresnel has demonstrated that this must be the case when the small displacement of a particle in the direction of any one of the co-ordinates produces forces in the direction of all, represented by multiples of that displacement. This is apparently the most general supposition that can be made. Mémoires de l'Institut, 1824. See also Griffin's Theory of Double Refraction.

We have spoken here of displacements as if the forces concerned in the transmission of a wave were thus put in play by absolute displacements. It is however plain from (103) that the forces on A really put in play are produced by relative displacements: but it is evident that these forces are the same as those that would be put in play by the absolute displacement

In like manner, when the direction of displacement is any whatever, the quantity (2u-u-u,) in its proper direction may be resolved into the direction of the co-ordinates, and the forces really acting on A will be the forces corresponding to these spaces considered as absolute displacements.